An extensive analysis of the presence of different altmetric indicators provided by Altmetric.com across scientific fields is presented, particularly focusing on their relationship with citations. Our results confirm that the presence and density of social media altmetric counts are still very low and not very frequent among scientific publications, with 15%-24% of the publications presenting some altmetric activity and concentrating in the most recent publications, although their presence is increasing over time. Publications from the social sciences, humanities and the medical and life sciences show the highest presence of altmetrics, indicating their potential value and interest for these fields. The analysis of the relationships between altmetrics and citations confirms previous claims of positive correlations but relatively weak, thus supporting the idea that altmetrics do not reflect the same concept of impact as citations. Also, altmetric counts do not always present a better filtering of highly cited publications than journal citation scores. Altmetrics scores (particularly mentions in blogs) are able to identify highly cited publications with higher levels of precision than journal citation scores (JCS), but they have a lower level of recall. The value of altmetrics as a complementary tool of citation analysis is highlighted, although more research is suggested to disentangle the potential meaning and value of altmetric indicators for research evaluation.